Problem Description
You are given an integer array coins representing coins of different denominations and an integer amount representing a total amount of money. Return the number of combinations that make up that amount. If that amount of money cannot be made up by any combination of the coins, return 0. You may assume that you have an infinite number of each kind of coin. The answer is guaranteed to fit into a signed 32-bit integer.
Key Insights
- The problem requires finding the number of combinations of coins that sum up to a given amount.
- Each coin can be used multiple times, which makes it a classic "unbounded knapsack" or "coin change" problem.
- Dynamic programming is an effective approach to solve this problem efficiently.
Space and Time Complexity
Time Complexity: O(n * amount), where n is the number of coins and amount is the target amount. Space Complexity: O(amount), which is used for the dynamic programming array to store combinations.
Solution
The solution utilizes a dynamic programming approach. We maintain a 1D array dp where dp[i] represents the number of combinations to achieve the amount i. We initialize dp[0] to 1 because there is one way to make the amount 0 (using no coins). For each coin in the coins array, we iterate through the dp array from the value of the coin to the target amount. For each value, we update dp[j] by adding the value of dp[j - coin], which signifies the number of ways to make the amount j including the current coin.